发表机构
Dalian University of Technology; National Yang Ming Chiao Tung University（大连理工大学; 国立阳明交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明四次Salem数对应的等权重Bernoulli卷积的关联维数严格小于1，通过碰撞概率与四维环面积分建立联系，并给出两个具体数值界。
AI 中文摘要
对于每一个四次Salem数$\beta\in(1,2)$，我们证明等权重的Bernoulli卷积$\nu_{\beta^{-1}}$的关联维数严格小于1。特别地，这两个对应的测度均不具有$L^2$密度。我们将关联维数与精确碰撞概率联系起来，并将这些概率表示为与定义多项式相关的四维环面上的正积分。单位圆共轭产生一个持续的振荡项；利用相应旋转的精心选择的返回时间，我们取差分使该项变小，同时控制所得构型的长度和代价。这些构型的许多良好分离的放置使得碰撞概率超过临界尺度，从而产生严格的维数下降。该定性论证不依赖于计算机辅助估计。独立的计算机辅助论证给出\\[ 0.99999<D_2(\nu_{\beta_1^{-1}})<1-10^{-65}, \qquad 0.99999<D_2(\nu_{\beta_2^{-1}})<1-10^{-75}, \\] 其中$\beta_1<\beta_2$是$(1,2)$中的两个四次Salem数。
英文摘要
For every quartic Salem number $β\in(1,2)$, we prove that the Bernoulli convolution $ν_{β^{-1}}$ with equal weights has correlation dimension strictly less than one. In particular, neither of the two corresponding measures has an $L^2$ density. We relate correlation dimension to exact collision probabilities and express these probabilities as positive integrals over a four-dimensional torus associated with the defining polynomial. The unit-circle conjugates produce a persistent oscillatory term; using carefully chosen return times of the corresponding rotation, we take differences that make this term small while controlling the length and cost of the resulting configurations. Many well-separated placements of these configurations then give a collision probability above the critical scale, which yields the strict dimension drop. The qualitative argument is independent of the computer-assisted estimates. Separate computer-assisted arguments give \[ 0.99999<D_2(ν_{β_1^{-1}})<1-10^{-65}, \qquad 0.99999<D_2(ν_{β_2^{-1}})<1-10^{-75}, \] where $β_1<β_2$ are the two quartic Salem numbers in $(1,2)$.
